Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33 https://doi.org/10.1007/s40430-018-0969-z (0123456789().,-volV)(0123456789(). ,- volV)
TECHNICAL PAPER
The effect of conductive baffles on natural convection in a power-law fluid-filled square cavity Afrasiab Raisi1 Received: 10 August 2017 / Accepted: 6 January 2018 Ó The Brazilian Society of Mechanical Sciences and Engineering 2018
Abstract This research involves the numerical study of the conjugate natural convection of a power-law fluid in a two-dimensional square cavity with a pair of conducting baffles. The lower wall of the cavity contains a heat source with constant heat flux, while a pair of conductive baffles is embedded on its upper wall. The left and right side walls and some parts of the lower wall of the cavity are thermally insulated and its upper wall is kept at a low temperature. The governing equations along with the corresponding boundary conditions are solved using the numerical finite difference method based on the control volume formulation and SIMPLE algorithm. The effects of pertinent parameters on flow and temperature fields and heat transfer rate are investigated. From the results of the numerical solution, increasing the Rayleigh number and baffles thermal conductivity enhance the thermal performance of the cavity. So that for Ra ¼ 106 ; with an increase in the thermal conductivity ratio from 1 to 100; the average Nusselt number increases by about 14% for both cases n ¼ 0:6 and n ¼ 1:2. As well as a decrease in power-law index, except in low Rayleigh numbers, increases the heat transfer rate. Based on the results, the increase of the average Nusselt number is about 283% for a power-law index reduction in the range of 0:6 n 1:2 at Ra ¼ 106 . Also, an increase of the length of the baffles deteriorates the thermal performance of the cavity at high Rayleigh numbers, while enhances it at low Rayleigh numbers. Keywords Natural convection Cavity Power-law Heat source Conductive baffles List of symbols b Length of baffles, m B Dimensionless length of baffles, ðb=lÞ d Distance between baffles, m D Dimensionless distance between baffles, ðd=lÞ Dij Rate of strain tensor, s-1 g Gravitational acceleration, m s-2 h Convection heat transfer coefficient, W m-2 K-1 k Fluid thermal conductivity, W m-1 K-1 kb Baffles thermal conductivity, W m-1 K-1 k Ratio of thermal conductivity ðkb =kÞ K Consistency coefficient l Cavity length, m n Power-law index Nu Local Nusselt number Technical Editor: Cezar Negrao. & Afrasiab Raisi
[email protected] 1
Engineering Faculty, Shahrekord University, PO Box 115, Shahrekord, Iran
Num p p P Pr q00 Ra s S T u; v U; V w W x; y X; Y z Z
Average Nusselt number Fluid pressure, Pa Modified pressure ðp þ qgyÞ Dimensionless pressure ðpl2 =qa2 Þ Prandtl number ðKl2n2 =qa2n Þ Heat flux, W m-2 Rayleigh number ðgbDTl2nþ1 =ðan K=qÞÞ Distance of heat source from the left wall, m Dimensionless distance of heat source from the left wall ðs=lÞ Temperature, K Velocity components in x, y directions, m s-1 Dimensionless velocity components ðul=a; vl=aÞ Length of the heat source, m Dimensionless length of the heat source ðw=lÞ Cartesian coordinates, m Dimensionless coordinates ðx=l; y=lÞ Thickness of baffles, m Dimensionless thickness of baffles, ðz=lÞ
Greek symbols a Thermal diffusivity, m2 s-1
123
33
b DT sij h l la q
Page 2 of 15
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
Thermal expansion coefficient, K-1 Reference temperature difference ðq00 l=kÞ, K Stress tensor, N m-2 Dimensionless temperature ððT Tc Þ=DTÞ Dynamic viscosity, N s m-2 Dimensionless apparent viscosity Density, kg m-3
Subscripts a Apparent c Cold m Average s Surface of the heat source
1 Introduction Natural convection is one of the economical and readily available heat transfer methods in many systems, including closed cavities, and has been well recognized by researchers in the past [1–5]. The rate of heat transfer within the enclosure can be enhanced by embedding conductive baffles on the walls of the enclosure that are active in the process of heat transfer. Sun and Emery [6] investigated the effect of the presence of a heat conductive baffle located vertically in the middle of a square cavity on the flow field and the heat transfer rate. They found that the length of the baffle and the ratio of its thermal conductivity to thermal conductivity of the fluid within the cavity (air) had a significant effect on the thermal performance of the cavity. Khorasanizadeh et al. [7] numerically investigated the free convection heat transfer and entropy generation of Cu–water nano-fluid within a cavity containing a heat conductive baffle on the hot lower wall. By using Newtonian model for nano-fluid viscosity, it was shown that the average Nusselt number increases with increasing the Rayleigh number and the volume fraction of nano-particles, regardless of the position of the baffle. In low Rayleigh numbers having conduction as the dominant mechanism of heat transfer, a movement of the baffle towards midsection of the cavity resulted in moderate conductivity and decreased the average Nusselt number. Whereas, for high Rayleigh numbers having convection as the main mechanism of heat transfer, movement of the baffle towards the midsection of the cavity resulted in a reinforced convection and an increased average Nusselt number. Hussain et al. [8] numerically investigated the effect of presence of a central tilted baffle on natural convection in a square air-filled cavity with the side wall being considered as corrugated. In that study, a heat source with constant heat flux was embedded on the bottom wall of the cavity, while the side walls were at constant temperature. The upper wall and the rest parts of the lower
123
walls were thermally insulated and there was an adiabatic tilted baffle in the center of the cavity. From the results, the presence of the tilted baffle and an increased corrugation frequency of the walls, especially for Gr [ 106 , had a considerable effect on the average Nusselt number, streamlines and isotherms. Also, compared to the tilted baffle, a baffle located in the vertical position resulted a greater average Nusselt number and behaved inversely when the baffle located in the horizontal position. Saravanan and Vidhya kumar [9] studied natural convection in a square cavity filled with air in the presence of two thin heat generating baffles. The horizontal walls of the cavity were insulated and different thermal boundary conditions were applied to its vertical walls. Baffles were at distance of D from each other, and were placed vertically in the center of cavity. They showed that the distance between baffles had significant effects on the flow pattern and heat transfer rate. Also, depending on the value of D, the inner circulating cells formed around the baffles either decreased or increased the strength of the main circulating cell. In many engineering applications, all enclosed fluids fall in the non-Newtonian fluids category. Food processing, polymer engineering, petroleum drilling, geophysical systems and cooling systems of electronic components are among examples which demonstrate natural convection of non-Newtonian fluids. One of the oldest researches in the field of natural convection in non-Newtonian fluids is the study by Ozoe and Churchill [10]. It involved the investigation of natural convection of both non-Newtonian Ostwald–de Waele (power-law) and Ellis fluids in a shallow horizontal cavity heated from below and cooled from above, and it was shown that the critical Rayleigh number for the onset of natural convection increased by increasing the flow behavior index. Kim et al. [11] studied transient natural convection of non-Newtonian power-law fluid in a vertical cavity through numerical method and scaling analysis. Horizontal walls of the cavity were considered as insulated and the simultaneous change of temperature in the vertical walls was responsible for the buoyancy force and natural convection in the cavity. Their findings showed that for high Rayleigh numbers and average Prandtl numbers, a decrease in power-law index strengthened the natural convection and hence enhanced the heat transfer rate. Also, the rheological property had significant effects on the transient and steady state processes. This feature is made pronounced by increasing the Rayleigh number and decreasing the Prandtl number. Lamsaadi et al. [12] investigated the natural convection of non-Newtonian power-law fluid in a shallow cavity using a combination of numerical and analytical methods. The horizontal walls of the cavity were tall and insulated while the vertical walls
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
were short and received steady cooling and heating with constant heat flux. From the results, the heat transfer and flow characteristics were not sensitive to an increase of the Prandtl number and aspect ratio for large enough values of these parameters. Thus, for non-Newtonian fluids with a large Prandtl number in shallow cavities, the effective parameters in flow field and heat transfer are n and Ra. Following this research, Lamsaadi et al. [13] in another study investigated the effect of the rotation of the considered cavity on flow field and heat transfer rate. From the study, for a given Rayleigh number, the rotation of the cavity had a significant effect on the rate of heat transfer. The maximum heat transfer occurred when the cavity was heated from below, and rotation effect of the cavity increased with decrease of value of n. Natural convection of power-law fluids in square and rectangular cavities with horizontal insulated walls and vertical walls under different boundary conditions in recent years has been well regarded by researchers [14–16]. The results of these studies showed that for rectangular cavities, where vertical walls are located at constant but different temperatures, variations of average Nusselt number with increasing aspect ratios did not demonstrate any uniform trend, while application of a constant heat flux boundary condition to vertical walls resulted in a uniform variation. The results also demonstrated the effect of Rayleigh number and power index on the flow field and heat transfer rate. Kefayati [17] studied entropy generation and heat and mass transfer by natural convection of power-law fluids in a square cavity using numerical finite difference lattice Boltzmann method. From the results, a decrease in powerlaw index resulted in a drop in the heat and mass transfer, while an increase in the Rayleigh number increased the heat and mass transfer and the entropy generation caused by fluid friction. In recent years, Kefayati conducted several studies on natural convection and also mixed convection of non-Newtonian fluids (nonofluids and polymers) using numerical methods FDLBM [18–20]. Alloui and Vasseur [21] examined numerically, the natural convection of a shear-thinning non-Newtonian fluid in a vertical enclosure using the Carreau–Yasuda model. From their study, a decrease in power-law index resulted in an increase in both the convection strength and heat transfer rate, and at low Rayleigh numbers the variations of n had little or no effect on the heat transfer rate. Cavities with insulated vertical walls, with heat applied from below and cooled from above is another class of cavities used in engineering applications. In such these configurations, Rayleigh number should be large enough to start a natural convection. Lamsaadi et al. [22] investigated numerically and analytically natural convection in a narrow rectangular horizontal cavity heated from below and cooled from above. From their findings, fluid flow, temperature
Page 3 of 15 33
distribution and heat transfer rate are sensitive to powerlaw index, but not to Prandtl number, especially for large Prandtl numbers (Pr C 100). In comparison with Newtonian fluid, natural convection in the shear thinning nonNewtonian fluid began in lower Rayleigh numbers, while in shear thickening non-Newtonian fluid it began in higher Rayleigh numbers. Mahrood et al. [23] experimentally investigated natural convection of non-Newtonian nanofluids in a cavity heated from below with a constant heat flux and cooled from above using Al2O3 and TiO2 nanoparticles and a 0.5 wt% aqueous solution of carboxymethyl cellulose as base fluid for preparing nanofluid. The natural convection heat transfer of non-Newtonian nanofluids was seen to be enhanced in low concentrations of nanoparticles, and an increase in volume fraction of particles above certain amounts resulted in a reduction in heat transfer rate. An optimum volume percent of 0.2 and 0.1 for Al2O3 and TiO2 nanoparticles, respectively was reported. Khezzar et al. [24] numerically investigated natural convection of non-Newtonian power-law fluids in a tilted square cavity heated from below and cooled from above. The study was conducted for different angles 0 B u B 90 and the heat transfer rate and flow field for different values of the aspect ratio, Rayleigh number and power-law index were examined. This study also confirmed previous studies results for the average Nusselt number changes with the power law index. Turan et al. [25] numerically studied laminar natural convection of power-law fluids in the square cavity heated by a constant heat flux from below and cooled from above. From the results, the Rayleigh number and n (power-law index) had significant effects on average Nusselt number, however, the Prandtl number had no significant effect on average Nusselt number. In the same manner, a comparison between the cavities heated from below and those heated from side indicated a higher average Nusselt number for cavities heated from the side than those heated from below for shear-thinning fluids when the Rayleigh numbers are high. But almost identical Nusselt number was observed for Newtonian and shearthickening fluids for both cases. Sairamu and Chhabra [26] numerically investigated natural convection flow from a square tilted cylinder to a non-Newtonian power-law fluid enclosed in a square cavity. The horizontal walls of cavity were insulated and vertical walls were at a constant temperature denoted by Tc , while square cylinder walls were at a constant temperature denoted by Th or at a constant heat flux. From the results, with low Grashof numbers and when the cylinder was at the centre of the cavity, two circulating cells were formed at the top and bottom of the square cylinder, while high Grashof numbers resulted in the merging of the circular cells. Also the rate of heat transfer was related to the
123
33
Page 4 of 15
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
Grashof number, the Prandtl number ð0:71 Pr 100Þ, the power-law index and position of the square cylinder. Baffles embedded in the enclosures (for example, electronic components installed inside the enclosures) have two different effects on flow field and heat transfer. First, they suppress the natural convection flow and, therefore, deteriorate the heat transfer rate. Second, with a thermal conductivity greater than that of the fluid enclosed in the cavity, they can improve heat transfer rate. Since the viscosity of power-law fluids is highly dependent on shear rates, for cavity containing power-law fluids, due to the drastic changes in shearing rate around baffles, the effects of baffles on natural convection becomes more important. To the best knowledge of the author, no studies in this area have been reported in the literature. Therefore, this study focuses on the effects of a pair of heat conductive baffles on natural convection flow and heat transfer, and cooling rate of a thermal source placed at the bottom of a square cavity filled with power-law fluid.
2 Problem description Figure 1 describes a schematic diagram of the two-dimensional square cavity under study in this research. The cavity is filled with a non-Newtonian power law fluid. A heat source with constant heat flux is placed on the lower wall of cavity, with a pair of conductive baffles symmetrically attached to the upper wall of the cavity. The distance between the baffles D ¼ d=l ¼ 0:2, the thickness of baffles Z ¼ z=l ¼ 0:05 and the length of the heat source
W ¼ w=l ¼ 0:4 are considered constant. Side walls of the cavity and the remaining parts of its lower wall are considered insulated and its upper wall is kept at a low temperature Tc . Apart from viscosity which is a function of the shear rate and density whose variations in buoyancy force is determined using Boussinesq approximation, the remaining thermo-physical properties of non-Newtonian power-law fluid are assumed to be constant. It is assumed that the flow is steady, laminar and incompressible. The Prandtl number is assumed to be 100 throughout the study.
3 Mathematical formulation Under the above mentioned assumptions, the equations that govern the conservation of mass, momentum and energy are as follows: ou ov þ ¼ 0; ox oy
ð1Þ
ou ou 1 op 1 osxx osxy þv ¼ þ þ ; ð2Þ ox oy q ox q ox oy ov ov 1 op 1 osxy osyy þ þ þ gbðT Tc Þ; u þv ¼ ox oy q oy q ox oy
u
ð3Þ 2 oT oT o T o2 T þv ¼a þ ; u ox oy ox2 oy2 o oT o oT kb kb þ ¼ 0: ox ox oy oy
ð4Þ ð5Þ
The last is the energy equation for the solid baffles. For a non-Newtonian fluid which follows the power-law model the viscous stress tensor is given by Andersson and Irgens [27]: oui ouj sij ¼ 2la Dij ¼ la ; ð6Þ þ oxj oxi where Dij is the rate of strain tensor for the two dimensional Cartesian coordinate and la is the apparent viscosity that is derived for the two dimensional Cartesian coordinate as [22]: ( " # )n1 2 ou 2 ov 2 ov ou 2 þ la ¼ K 2 þ ; ð7Þ þ ox oy ox oy
Fig. 1 Schematic diagram, coordinates and boundary conditions of the physical model
123
where K and n are power-law model constants. K is the consistency coefficient and n is the power-law index. Where, n\1 is for shear-thinning fluids and n [ 1 is for shearthickening fluids. When n ¼ 1, a Newtonian fluid is obtained. The above governing equations can be written in nondimensional forms using the following non-dimensional parameters [28, 29]:
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
y ul Y¼ ; U¼ ; l a T Tc q00 l h¼ ; DT ¼ ; DT k
x X¼ ; l
vl ; a kb k ¼ : k V¼
P¼
pl2 ; qa2
The non-dimensional equations are then given by [29]: ð9Þ
ð10Þ
oV oV oP þV ¼ oX oY oY o oV o oU oV þ Pr l þ la þ2 þ RaPrh; oX a oY oX oY oY
U
ð11Þ oh oh o2 h o2 h þV ¼ þ ; U oX oY oX 2 oY 2 o oh o oh k k þ ¼ 0; oX oX oY oY
ð12Þ ð13Þ
where, la , is the dimensionless apparent viscosity and is defined as: ( " # )n1 2 oU 2 oV 2 oV oU 2 þ þ ; ð14Þ la ¼ 2 þ oX oY oX oY Given that the viscosity of power-law fluids depends on shear rate, the Prandtl and Rayleigh numbers for these fluids also vary with the shear rate. But, in numerical studies the nominal Prandtl and Rayleigh numbers are used according to Eq. (15): Ra ¼
gbDTl2nþ1 ; an K=q
Pr ¼
Kl2n2 : qa2n
oh ¼ 0 for X ¼ 0; 1 oX and 0 Y 1 U ¼ V ¼ h ¼ 0 for Y ¼ 1 and 0 X 1 oh ¼ 0 for Y ¼ 0 U¼V ¼ 8 oY 1 > > > < 0 X\ S 2 W and > 1 > > : S þ W \X 1 2 oh U ¼ V ¼ 0 and ¼ 1 for Y ¼ 0 oY 1 X Sþ W ; 2 U¼V ¼
ð8Þ
oU oV þ ¼ 0; oX oY oU oU oP þV ¼ U oX oY oX o o oV oU oU þ Pr 2 l la þ þ ; oX a oX oY oY oX
Page 5 of 15 33
ð16Þ
and
1 S W 2
In addition to the no-slip boundary condition, thermal boundary conditions at the solid–fluid interfaces of the baffles are also expressed based on the continuation of the temperatures and heat fluxes. Hence, the boundary conditions at the interfaces are as follows: U ¼ V ¼ 0 and oh oh hfluid ¼ hsolid ; ¼ K : on fluid on solid
ð17Þ
After solving the governing equations numerically, as a measure of the heat transfer rate, the local Nusselt number on the heat source surface can be defined as follows: Nu ¼
hl q00 l ¼ : k ðTs Tc Þk
ð18Þ
In Eq. (16), h is the convection heat transfer coefficient. Using the dimensionless parameters, the following relationship is obtained for the local Nusselt number: Nu ¼
1 ; hs ð X Þ
ð19Þ
where hs is the dimensionless temperature of the heat source. The average Nusselt number can be obtained by integrating the local Nusselt number along the heat source.
ð15Þ
The non-dimensional boundary conditions, used to solve the non-dimensional governing equations are follows:
5.68 5.64
Num
5.60 5.56
120x120
0.23 0.22
θs, max
0.21 0.20 0
5000
10000
15000
20000
25000
Grid points
Fig. 2 Grid independency study ðRa ¼ 105 ; k ¼ 1; n ¼ 0:8Þ
123
33
Page 6 of 15
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
Table 1 Comparison of the average Nusselt number (Num ) extract from present code and Turan et al. [25]
Ra ¼ 6 104
n
Ra ¼ 105
Ref. [25]
Present work
Error (%)
Ref. [25]
Present work
Error (%)
0.8
3.8666
3.8395
0.7
4.2167
4.2431
0.62
0.9
3.4500
3.4943
1.28
3.7667
3.7547
0.32
1
3.1166
3.1070
0.3
3.3833
3.3740
0.27
1.2
2.5500
2.5342
0.62
2.7833
2.7749
0.3
1.4
2.1000
2.0974
0.12
2.3167
2.3205
0.16
0.2 Present Work Turan et al. [25]
n = 0.8, 1, 1.2
0.1
θX = 0.5 0.0
-0.1
-0.2 0.0
0.2
0.4
0.6
0.8
1.0
Y
Fig. 3 Validation of the present study against the results of Turan et al. [25] for Ra ¼ 105 Sþ12W
Z
1 Num ¼ W
1 dX hs ð X Þ
ð20Þ
S12W
4 Numerical method, grid study and validation Using the finite difference method based on the control volume formulation, the dimensionless Eqs. (9–13) with boundary conditions given in Eq. (15) are discretized. Power-law scheme is used to discretize the convection– diffusion terms. A staggered grid system in which the velocity components are calculated on control volume faces is used. Patankar’s SIMPLE algorithm [30] is applied in solving discrete equations simultaneously and coupling the velocity and pressure fields. The governing equations for fluid and solid baffles are simultaneously solved in a single computational domain. A very large number is assigned to the apparent viscosity of gride points laying in the solid baffles. This results in a zero velocity throughout these regions. The thermal boundary condition is established at the internal interfaces by taking into account the continuation of temperatures and heat fluxes. The
123
convergence criterion is to reduce the maximum mass residual of the grid control volume below 107 . To find a grid with appropriate number of points, such that the results of the numerical solution are independent of the number of grid points, the computer code was run for a grid series with the number of points from 20 20 to 140 140 for Ra ¼ 105 , n ¼ 0:8, k ¼ 1 and B ¼ 0:2. In Fig. 2, the variations of the average Nusselt number Num and heat source maximum temperature hs;max with the number of grid points are shown. According to Fig. 2, a greater than 120 120 number of grid points result in a slight change in the parameters. Thus, a non-uniform grid of 120 120 having smaller meshes inside baffles areas is used. For validation of numerical code, according to the research conducted by Turan et al. [25], a square cavity containing a power-law fluid is considered. The side walls of the cavity are thermally insulated, whereas its horizontal walls are exposed to a constant heat flux. Table 1 and Fig. 3 illustrate a comparison of the results of present numerical simulation with the results of Turan et al. [25], in different conditions. As seen, there is very good consistency between the results of the numerical solution and the results provided by Turan et al.
5 Results The results of this research are provided in form of the effects of the Rayleigh number 103 Ra 106 , the powerlaw Index 0:6 n 1:4, the baffles thermal conductivity ðk ¼ 1; 10; 100Þ and baffles length 0 B 0:4 on flow and temperature fields and heat transfer rate. In this study, Prandtl number, length and position of the heat source, and also the thickness and position of the baffles are considered constant.
5.1 Effects of Rayleigh number and power-law index In this part of the study, the effects of Ra and n on flow field and thermal performance of the cavity are investigated. It is assumed that the length of the baffles and their
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
θs,max = 0.5009
n=1
θs,max = 0.5012
0.031
4
θs,max = 0.4988
0. 2 80
6
θs,max = 0.2195
0.0941 0.1568
0.2 0.
34
82
0.2195 0. 28 22
0. 3
4
θs,max = 0.2916
44 9
θs,max = 0.3712
2.3366
0.146 0.128
4.1911
0.0 96
0.110
0.162
39
θs,max = 0.2232
0.084
0. 0 98
7.7012
0.064
13.2407
0.063
0.072
24.372
5 0.07
0.112
Ra = 10
6
θs,max = 0.1710
0. 1
0. 11 6
0.186 9 0.20
9
θs,max = 0.1436
0.10
7.5556
5
91 0.0
0.082
Ra = 10
0.0314
0.156 0.219
44
θs,max = 0.5016
0.094
0.2155
0.3911
Ra = 10
4
34 30
0.282
0. 3
0.031
0.1559
0.
0.219
θs,max = 0.5006
0.0935
83
0.219
34 5
0.0312
0.21
0.156
0.282
0.
0.094
0.157
0.0177
0.0177
0.282
0.031
0.094
0.219
0.0149
Ra = 10
3
0.094
0.3 4
θs,max = 0.5007
0.031
0.157
0.0149
n = 1.2
0.1563
n = 0.8
Page 7 of 15 33
Fig. 4 Streamlines (left) and isotherms (right) for different Rayleigh numbers and power-law indices ðPr ¼ 100; k ¼ 1; B ¼ 0:2Þ
thermal conductivity are constant (B ¼ 0:2; K ¼ 1). Figure 4 shows streamlines and isotherms for different Rayleigh numbers at three different values of power-law index (n ¼ 0:8; 1 and 1:2). Due to the symmetry of the physical model compared to the vertical mid section of the cavity, streamlines are shown in the left half and isotherms in the
right half of the cavity. In general, an increase in the Rayleigh number, results in a corresponding rise in buoyancy force thereby enhancing the natural convection. The apparent viscisity of shear thinning non-Newtonian fluids (power-law fluids with n\1) has a large amount at small shear rates and it reduces by increasing the shear rate. But,
123
33
Page 8 of 15
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
(a)
(a) n = 0.8
60
14
Ra = 10 5
n = 0.8 n=1 n = 1.2
40
Ra = 104
12
Ra = 105 Ra = 106
10
20
V Y= 0.5 0
Nu
8 6
-20
4 -40
2 -60 0.0
0.2
0.4
0.6
0.8
1.0
0 0.3
X
0.4
0.5
0.6
0.7
0.6
0.7
X
(b)
(b) n = 1.2
0.24
9 n = 0.8 n=1 n = 1.2
0.20
Ra = 104
8
Ra = 10
5
Ra = 106
7 0.16
6
Nu
θY= 0.5 0.12
5 4
0.08
3 2
0.04 0.0
0.2
0.4
0.6
0.8
1.0
X
Fig. 5 Variation of vertical velocity component (a) and temperature (b) along the mid-section of the enclosure for Ra ¼ 105 and different power-law indices ðPr ¼ 100; k ¼ 1; B ¼ 0:2Þ
this behavior occurs inversely for shear thickening nonNewtonian fluids (power-law fluids with n [ 1) [31]. Therefore, a low shear rate accounts for a decrease in the apparent viscosity with increasing power-law index, resulting in a strengthened natural convection, whereas, a high shear rate accounts for an increase in the apparent viscosity with a rise in the power-law index and a weakened natural convection. In all cases, two symmetrical vortices are formed within the cavity whose strength increase with increasing Rayleigh number. For Ra ¼ 103 with a low shearing rate, the strength of vortices is increased with an increase in power-law index. For Ra ¼ 104 ;105 and 106 the shear rate increases, hence the strength of the vortices decreases with increasing powerlaw index. Baffles embedded in the cavity resulted in the deviation of vortices direction and a decrease in the strength of vortices. In addition, they also prevent the fluid stream in the vicinity of the upper wall. Considering
123
1 0.3
0.4
0.5
X
Fig. 6 Variation of local Nusselt number along the heat source for different Rayleigh numbers a n ¼ 0:8 and b n ¼ 1:2, ðPr ¼ 100; k ¼ 1; B ¼ 0:2Þ
isotherms, values of Ra ¼ 103 and 104 resulted in a weak natural convection with isotherms almost horizontal and parallel to each other. This suggests conduction as a dominant mechanism of heat transfer in these condition. Thus, for Ra ¼ 103 and 104 because heat transfer rate is low, the heat source maximum temperature is high and does not change much with the Rayleigh number and power-law index. For Ra ¼ 105 and 106 with increasing strength of the vortices, natural convection is the main mechanism of heat transfer. In this case, the maximum temperature of the heat source decreases significantly with increase of Rayleigh number and decrease of power-law index. In Fig. 5a, b, the vertical component of dimensionless velocity and dimensionless temperature respectively are shown along the horizontal mid-section of the cavity for Ra ¼ 105 at three different values of power-law index (n ¼ 0:8;1 and 1:2). According to the direction of the
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
(a)
18 Ra = 104 Ra = 105 6 Ra = 10
16 14 12
Num
10 8 6 4 2 0 0.6
0.8
1.0
1.2
1.4
n
(b) 0.6 0.5 0.4
θs, max 0.3 0.2 Ra = 104 5 Ra = 10 Ra = 106
0.1 0.0 0.6
0.8
1.0
1.2
1.4
n
Fig. 7 Variation of average Nusselt number (a) and heat source maximum temperature (b) versus power-law index at various Rayleigh numbers ðPr ¼ 100; k ¼ 1; B ¼ 0:2Þ
vortex flows, it is seen that the velocity near the side walls is downwards and in the middle part of the cavity it is upwards. It is also shown that the absolute vertical velocity decreases with an increase in the power-law index. This is as a result of an increase in the apparent viscosity of the fluid with an increase in n. Reduction in fluid velocity causes less heat to be transferred out of the cavity. Therefore, fluid temperature is increased with increasing n. In Fig. 6, the effects of the Rayleigh number on the local Nusselt number along the heat source has been shown at two values of the power-law index (n ¼ 0:8 and 1:2). At Ra ¼ 104 , in which conduction is the main mechanism for heat transfer, local Nusselt number is almost constant along the heat source. According to the vortices circulation, direction of fluid flow in the vicinity of thermal source is from the sides toward the center of it. Therefore, on the sides of the heat source, the temperature difference between the fluid and heat source is high and the local
Page 9 of 15 33
Nusselt number is significant. With fluid movement toward the central line of heat source, the temperature difference reduces and the local Nusselt number decreases. The results also show that an increase of the Rayleigh number increases the heat transfer rate; and an increase of the power-law index suppresses the convective flows and reduces the rate of heat transfer. Figure 7a, b presents the variations of average Nusselt number and maximum temperature of the heat source, respectively, with the power-law index at different values of Rayleigh numbers. At Ra ¼ 104 , natural convection is weak and apparent viscosity variation due to the change of power-law index dose not have significant effect on thermal performance of cavity. Therefore, at Ra ¼ 104 , the average Nusselt number dose not change with power-law index. For Ra ¼ 105 ; 106 , in which natural convection is strengthened, increasing apparent viscosity due to increase of the power-law index weakens the natural convection. Thus, for Ra ¼ 105 ; 106 the average Nusselt number decreases with increasing n. At Ra ¼ 106 in the range of 0:8 n 1, the average Nusselt number decreases with a more slight slope. According to Fig. 4, it can be seen that the increase in power-law index in the range of 0:8 n 1, accounts for a significant reduction in the strength of vortices. Therefore, it is expected that the heat transfer rate is also significantly reduces. But for n ¼ 0:8, two secondary vortices are formed in the vicinity of baffles. These secondary vortices prevent contact of primary vortices with the upper cold wall. When power-law index increases from 0.8 to a value of 1, two phenomena occur simultaneously. First, strength of the primary vortices is reduced which causes the average Nusselt number to decrease. Second, secondary vortices are destroyed and primary vortices can have contact with cool upper wall and increase the heat transfer rate. The interactive effects between these two phenomena cause the average Nusselt number in this range to decrease with a more slight slope. The maximum temperature of the heat source with changes in Ra and n is opposite of the average Nusselt number.
5.2 Effects of thermal conductivity of baffles In this part of the study the effects of thermal conductivity ratio (K ¼ ks kf ) for n ¼ 0:8; 1 and Ra ¼ 103 ; 105 are presented. The length of the baffles are assumed to be constant ðB ¼ 0:2Þ. Figure 8a, b, illustrates the streamlines (left) and isotherms (right) for n ¼ 0:8 and n ¼ 1:2, respectively, at three different values of thermal conductivity ratio (K ¼ 1; 10 and 100) for Ra ¼ 103 and 105 . According to Fig. 6a, the thermal conductivity ratio has a little effect on the strength of the primary vortices. In some cases, a pair of
123
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
0.031
θ s,max = 0.4699
0.088
0.
34 4
θ s,max = 0.4544 0.028 0.085
0.147
0.219
0.282
*
k = 100
0.029
0.157
0.0149
Ra = 10
3
0.094
*
k = 10
-.0013
θ s,max = 0.5009
0.142
0.206
θ s,max = 0.2195
0.26
0. 3
0.199
0.0106
*
k=1
-0.0006
Page 10 of 15
0.0108
33
4
23
θ s,max = 0.2090
0.256 0.3 1
2
θ s,max = 0.1984 0. 0
74
0.087
0. 0
7.5360
91
0.078
0. 0
7.5456
6
0.110
5
0.0 9
0.062
7.4822
50
0.105
0.082
Ra = 10
0. 0
2 05 0.065
69 0. 0
0.
37
(a) n = 0.8 θ s,max = 0.5007
k*= 10
θ s,max = 0.4725
0.031
0.089
θ s,max = 0.3712
0.
0.26 6
θ s,max = 0.3479
0.023
0.25
θ s,max = 0.3339
0.022 0.06
0.021
5
0.0 63
0. 1 88
0.146
6 0. 19
0.152
2.2695
04
2
2.2958
0. 1
0.16
09 0.1
16
9 0.20
2.3366
7
0. 31 4
1 0.
5
0.200
32 5
0. 07 0
Ra = 10
0.086 0.143
0.207
0.0165
0.3 44
0.029
-0.004
0.148
0.219 0.282
θ s,max = 0.4562
0.030
-0.002
0.156
0.0177
Ra = 10
3
0.094
k*= 100
0.0158
k*= 1
(b) n = 1.2 Fig. 8 Streamlines (left) and isotherms (right) for Ra ¼ 103 ; 105 and different values of k a n ¼ 0:8 and b n ¼ 1:2, ðPr ¼ 100; B ¼ 0:2Þ
secondary vortices are formed in the vicinity of baffles. Secondary vortices are formed under the influence of two factors. First, the primary vortices should be weak, so that the fluid around the baffles is not affected by circulation of primary vortices flow. Second, thermal conductivity of the
123
baffles must be great and temperature gradient along the baffles low. In these conditions, the stagnant fluid between the baffles and left and right walls of the cavity are circulated under the influence of temperature difference of baffles and side walls, resulting in the formation of the
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
(a) n = 0.8
12 10
k *= 1 k *= 10 * k = 100
8
Num 6 4 2 0 103
104
105
106
Ra
(b) n = 1.2
7 k* = 1 k* = 10 k* = 100
6
to conduction heat transfer mechanism. Therefore, average Nusselt number increases with increase of thermal conductivity of baffles. Furthermore, as expected, with increasing Rayleigh number, natural convection is strengthened and the average Nusselt number increases. By comparing Fig. 9a, b, it is seen that this increase is more for n ¼ 0:8. This is because with the increasing Rayleigh number, shear rate increases and the viscosity of non-Newtonian shear thinning fluids decreases, resulting in easy movement of fluid under influence of Buoyancy force. Also, according to Fig. 9, it can be seen that the increasing effect of baffles thermal conductivity on average Nusselt number in high Rayleigh numbers is significant. This is because at high Rayleigh numbers, the main mechanism of heat transfer is natural convection and surrounding fluid around the cold baffles has higher temperature, and this temperature difference between cold baffles and hot fluid cause more heat to be transferred to the upper wall.
5.3 Effects of baffles length
5
Num 4 3 2 1 103
Page 11 of 15 33
104
105
106
Ra
Fig. 9 Variation of average Nusselt number with power-law index for different k , ðPr ¼ 100; B ¼ 0:2Þ
secondary vortices. The strength of the secondary vortices increases with an increase in thermal conductivity of the baffles. A comparison of Figs. 8a and 9b shows an increase in the strength of secondary vortices with increasing n. Given that secondary vortices are formed at Ra ¼ 103 and in this case the shear rate is low, with increase of n, apparent viscosity of the fluid is reduced and as a results strength of primary vortices increased. At Ra ¼ 106 , strength of primary vortices is high, thus the formation of secondary vortices is not possible. According to the isothermal lines, it can be seen that the increase in thermal conductivity of baffles enhances thermal performance of the enclosure and, therefore, reduces the maximum temperature of the heat source. In Fig. 9a, b, variations of average Nusselt number based on variations of Rayleigh number for three different values of thermal conductivity ðk ¼ 1; 10; 100Þ have been shown for n ¼ 0:8 and n ¼ 1:2, respectively. Baffles with a high thermal conductivity enhance thermal performance of the cavity due
In this section, the effects of a change in the length of baffles within the range 0 B 0:4, on flow and temperature fields and heat transfer rate for n ¼ 0:8; 1:2 at Ra ¼ 103 ; 105 are investigated. Thermal conductivity of baffles was assumed constant ðk ¼ 10Þ. Figure 10a, b shows the effects of the baffles length on the streamlines (left) and isotherms (right) at Ra ¼ 103 ; 105 for n ¼ 0:8 and n ¼ 1:2, respectively. As seen in Fig. 10, with increase of baffles length, strength of the primary vortices is reduced. Under favorable conditions, the secondary vortices are formed on the side of the baffles. Increasing length of the baffles increases the strength of the secondary vortices. At Ra ¼ 103 , having a low shear rate, with increasing powerlaw index, the viscosity of the fluid decreases and the strength of primary and secondary vortices increases. At Ra ¼ 105 , having an increased shear rate, with increasing power-law index, viscosity of fluid increases and strength of primary vortices decreases, resulting in the destruction of the secondary vortices. At Ra ¼ 105 , when n ¼ 1:2 and B ¼ 0:4, owing to the significant drop in strength of primary vortices, secondary vortices are formed. Baffles have two different effects on flow and temperature fields. First, they act as a barrier to the natural convection flow and thus deteriorate the thermal performance of the cavity, second when the thermal conductivity of baffles is more than thermal conductivity of the fluid ðk [ 1Þ, baffles transfer more heat to the upper wall by conduction and enhance thermal performance. According to Fig. 10, it is seen that for low Rayleigh numbers, with conduction as the dominant mechanism of heat transfer, baffles enhances thermal performance of the cavity, and an increase in the length of baffles decreases the maximum
123
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
0.094
0.088
0.0108
3
3
θ s,max = 0.2090
0.189
0.2 43
0.2 9
7
θ s,max = 0.2413
0. 0. 0 0 2 6 39
23
0. 0
91
0.0 91
5
60
0. 0
0. 0
78
0.105
0.046
7.5456
0.030
52 5 0.0 0.06
34 0.05 7
0.135
0.264 0.3 2
0. 0
0.068
0.081
0.136
0.0127
0.281
0. 0
8.2216
0.027
0.206
θ s,max = 0.1820
Ra = 10
θ s,max = 0.4313
0.147
0.218
0.3 4
B = 0.4
0.029
0.156
3
Ra = 10
θ s,max = 0.4675
-0.0006
0.031
B = 0.2
0.121
0.106
θ s,max = 0.499
-0.001
B=0
0.0086
Page 12 of 15
6.2308
33
(a) n = 0.8
0.148
θ s,max = 0.3266
0. 3
0.26 6
25
0.022
1
0.2 42
0.2 9
6
θ s,max = 0.3767 0.024
-0.0096
0.071 0.1 18
0. 2
65 0. 1
2.2958
9 10 12
84 0. 1
96 0. 1
0.152
2.4851
0.188
0.
5
0.135
0.065
2 0.10 0.143
Ra = 10
06
0.081
θ s,max = 0.3479
0.020 0.
0.027
0.207
0.0165
0. 3 41
θ s,max = 0.4306
-0.0023
0.089
0.217 0.279
B = 0.4
0.030
0.155
0.0161
Ra = 10
3
0.093
θs,max = 0.4725
0.0092
0.031
B = 0.2
1.6435
θ s,max = 0.4958
-0.0013
B=0
(b) n = 1.2 Fig. 10 Streamlines (left) and isotherms (right) for different baffles lengths and Ra ¼ 103 ; 105 a n ¼ 0:8 and b n ¼ 1:2 ðPr ¼ 100; k ¼ 10Þ
temperature of the heat source. Whereas, at high Rayleigh numbers, with natural convection as the main mechanism of heat transfer, baffles deteriorate the thermal performance of cavity, and an increase in the length of baffles results in an increase in the maximum temperature of the heat source.
123
To understand better how baffles affect flow and temperature fields, profiles of the vertical component of dimensionless velocity (left) and dimensionless temperature (right) are shown along the horizontal mid-section of the cavity for n ¼ 0:8 in Fig. 11. From the figure,
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33 0.06
Page 13 of 15 33 0.22
B=0 B = 0.2 B = 0.4
0.04
B=0 B = 0.2 B = 0.4
0.20
0.02
0.18
θ Y = 0.5
VY = 0.5
0.00
0.16
-0.02
0.14
-0.04
0.12 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
X
0.6
0.8
1.0
0.8
1.0
X
0.14
80 B=0 B = 0.2 B = 0.4
60
0.12
40
0.10
θ Y = 0.5
20
V Y = 0.5
0.08
0 0.06 B=0 B = 0.2 B = 0.4
-20 0.04
-40 0.02
-60 0.0
0.2
0.4
0.6
0.8
1.0
X
0.0
0.2
0.4
0.6
X
Fig. 11 Variation of vertical velocity component (left) and temperature (right) along the mid-section of the enclosure for n ¼ 0:8 at different baffles lengths ðPr ¼ 100; k ¼ 10Þ
increasing the length of baffles decreases the vertical components of velocity along mid-section of the cavity. At Ra ¼ 105 for B ¼ 0:4, the vertical component of velocity in the middle part of the cavity ð0:25 X 0:75Þ is almost a constant value. Also, according to the temperature profile, it can be seen that in Ra ¼ 103 , where the main mechanism of heat transfer is conduction, fluid temperature along the horizontal midsection of cavity drops with an increase of the baffle length, thus enhancing the thermal performance of the cavity. For Ra ¼ 105 where natural convection is the main mechanism of heat transfer, an increase in baffles length results in an increase in the temperature of the fluid in the horizontal mid sections of cavity thereby reducing the thermal performance of the cavity. Variations of average Nusselt number with Rayleigh number for three values of baffle length ðB ¼ 0; 0:2; 0:4Þ, and for n ¼ 0:8 and n ¼ 1:2 are respectively shown in Fig. 12a, b. According to Fig. 12, at low Rayleigh numbers, the average Nusselt number increases with increasing baffles length. Although baffles weaken natural convection, but since convection is weak at low Rayleigh numbers and the main
mechanism of heat transfer is conduction, therefore, the average Nusselt number increases with an increase of baffles length. This increase is more significant for n ¼ 1:2 compared to n ¼ 0:8. For n ¼ 1:2, due to increased fluid viscosity, conduction has greater role in heat transfer and therefore baffles have more effective role in increasing the heat transfer rate. In high Rayleigh numbers with natural convection as the main mechanism of heat transfer, an increase in the length of baffles reduces the strength of the vortices, hence the average Nusselt number decreases with increasing baffles length. In Fig. 12a, for high Rayleigh numbers ðRa ¼ 106 Þ average Nusselt number for B ¼ 0:2 has been significantly reduced, compared to B ¼ 0. This significant reduction is due to an increase in the heat transfer rate resulting from very strong natural convection within the cavity together with the contact of the circulating fluid with the upper cold wall of the cavity, in the absence of baffles and at high Rayleigh numbers and when n ¼ 0:8. However, the presence of baffles inside the cavity results firstly in a reduction of vortices strength and secondly the diversion of flow path and preventing the contact of circulating fluid with upper
123
33
Page 14 of 15
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
(a) n = 0.8
installed on its upper wall, is investigated numerically. The effects of variations of Rayleigh number, power-law index, thermal conductivity ratio and length of the baffles on flow and temperature fields and heat transfer rate have been investigated. According to the results of numerical solution, the following conclusions can be expressed.
14 B=0 B = 0.2 B = 0.4
12 10
Num 8
–
6 4
–
2 0 103
104
105
106
Ra
–
(b) n = 1.2 8 B=0 B = 0.2 B = 0.4
7
–
6 5
Num
4 3 2
–
1 103
104
105
106
Ra
In general, Increasing the Rayleigh number and decreasing the power-law index, enhance the natural convection inside the cavity, resulting in a better thermal performance of the cavity. Increased thermal conductivity of baffles increases the heat transfer rate from fluid to upper wall, especially at high Reynolds numbers. Baffles have two different effects on flow and temperature fields. First, they reduce the strength of primary vortices formed within the cavity, and secondly, if their thermal conductivity is more than thermal conductivity of fluid, rate of heat conduction from fluid to the upper wall is increased. At low Rayleigh numbers in which conduction is the main mechanism of heat transfer, thermal performance of the cavity enhances as the length of the baffles increases. Whereas, at high Rayleigh numbers in which natural convection is the main mechanism of heat transfer, increasing baffles length deteriorates thermal performance of the cavity. At a critical Rayleigh number Racr , average Nusselt number is not affected by the baffles length. With an increase of power-law index the critical Rayleigh number increases.
Fig. 12 Variation of average Nusselt number with Ra for different baffles lengths ðPr ¼ 100; k ¼ 10Þ
cold wall. Therefore, in this case, baffles significantly reduce the average Nusselt number. The results also show that at a critical Rayleigh number Racr the baffles length has no effect on the average Nusselt number. In fact, for Ra\Racr , the main mechanism of heat transfer is conduction and increase of baffles length results in higher values of average Nusselt number. Whereas, for Ra [ Racr , natural convection is the main mechanism of heat transfer and increase of baffles length results in lower values of average Nusselt number. The critical Rayleigh number is a function of the power-law index, so that increases as the power-law index increases.
6 Conclusions In this study, natural convection of non-Newtonian powerlaw fluids in a square cavity on whose lower wall there is a local heat source, and a pair of heat conductive baffles
123
References 1. Bejan A (2013) Convection heat transfer. Wiley, New York 2. Catton I (1978) Natural convection in enclosures. In: Proceedings of the sixth international heat transfer conference, pp 13–31 3. de Vahl Davis G (1983) Natural convection of air in a square cavity: a bench mark numerical solution. Int J Numer Methods Fluids 3(3):249–264 4. Ostrach S (1988) Natural convection in enclosures. J Heat Transf 110(4b):1175–1190 5. Padilla ELM, Lourenc¸o MA, Silveira-Neto A (2013) Natural convection inside cubical cavities: numerical solutions with two boundary conditions. J Braz Soc Mech Sci Eng 35(3):275–283 6. Sun YS, Emery AF (1997) Effects of wall conduction, internal heat sources and an internal baffle on natural convection heat transfer in a rectangular enclosure. Int J Heat Mass Transf 40(4):915–929. https://doi.org/10.1016/0017-9310(96)00120-2 7. Khorasanizadeh H, Amani J, Nikfar M (2012) Numerical investigation of Cu–water nanofluid natural convection and entropy generation within a cavity with an embedded conductive baffle. Scientia Iranica 19(6):1996–2003 8. Hussain SH, Jabbar MY, Mohamad AS (2011) Influence of presence of inclined centered baffle and corrugation frequency on
Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018)40:33
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
natural convection heat transfer flow of air inside a square enclosure with corrugated side walls. Int J Therm Sci 50(9):1799–1808 Saravanan S, Vidhya kumar AR (2014) Natural convection in square cavity with heat generating baffles. Appl Math Comput 244:1–9. https://doi.org/10.1016/j.amc.2014.06.092 Ozoe H, Churchill SW (1972) Hydrodynamic stability and natural convection in Ostwald–de Waele and Ellis fluids: the development of a numerical solution. Am Inst Chem Eng 18(6):1196–1207 Kim GB, Hyun JM, Kwak HS (2003) Transient buoyant convection of a power-law non-Newtonian fluid in an enclosure. Int J Heat Mass Transf 46(19):3605–3617 Lamsaadi M, Naimi M, Hasnaoui M (2006) Natural convection heat transfer in shallow horizontal rectangular enclosures uniformly heated from the side and filled with non-Newtonian power law fluids. Energy Convers Manag 47(15):2535–2551 Lamsaadi M, Naimi M, Hasnaoui M, Mamou M (2006) Natural convection in a tilted rectangular slot containing non-Newtonian power-law fluids and subject to a longitudinal thermal gradient. Numer Heat Transf Part A Appl 50(6):561–583 Ternik P, Rudolf R (2013) Laminar natural convection of nonNewtonian nanofluids in a square enclosure with differentially heated side walls. Int J Simul Model 12(1):5–16 Turan O, Sachdeva A, Chakraborty N, Poole RJ (2011) Laminar natural convection of power-law fluids in a square enclosure with differentially heated side walls subjected to constant temperatures. J Nonnewton Fluid Mech 166(17):1049–1063 Turan O, Sachdeva A, Poole RJ, Chakraborty N (2013) Aspect ratio and boundary conditions effects on laminar natural convection of power-law fluids in a rectangular enclosure with differentially heated side walls. Int J Heat Mass Transf 60:722–738 Kefayati GR (2015) FDLBM simulation of entropy generation in double diffusive natural convection of power-law fluids in an enclosure with Soret and Dufour effects. Int J Heat Mass Transf 89:267–290 Kefayati GR (2014) Simulation of non-Newtonian molten polymer on natural convection in a sinusoidal heated cavity using FDLBM. J Mol Liq 195:165–174
Page 15 of 15 33
19. Kefayati GR (2015) FDLBM simulation of mixed convection in a lid-driven cavity filled with non-Newtonian nanofluid in the presence of magnetic field. Int J Therm Sci 95:29–46 20. Kefayati GR (2015) Mesoscopic simulation of mixed convection on non-Newtonian nanofluids in a two sided lid-driven enclosure. Adv Powder Technol 26(2):576–588 21. Alloui Z, Vasseur P (2015) Natural convection of Carreau–Yasuda non-Newtonian fluids in a vertical cavity heated from the sides. Int J Heat Mass Transf 84:912–924 22. Lamsaadi M, Naimi M, Hasnaoui M (2005) Natural convection of non-Newtonian power law fluids in a shallow horizontal rectangular cavity uniformly heated from below. Heat Mass Transf 41(3):239–249 23. Mahrood MRK, Etemad SG, Bagheri R (2011) Free convection heat transfer of non Newtonian nanofluids under constant heat flux condition. Int Commun Heat Mass Transf 38(10):1449–1454 24. Khezzar L, Siginer D, Vinogradov I (2012) Natural convection of power law fluids in inclined cavities. Int J Therm Sci 53:8–17 25. Turan O, Lai J, Poole RJ, Chakraborty N (2013) Laminar natural convection of power-law fluids in a square enclosure submitted from below to a uniform heat flux density. J Nonnewton Fluid Mech 199:80–95 26. Sairamu M, Chhabra R (2013) Natural convection in power-law fluids from a tilted square in an enclosure. Int J Heat Mass Transf 56(1):319–339 27. Andersson H, Irgens F (1990) Film flow of power-law fluids. Encycl Fluid Mech 9:617–648 28. Saha S, Sultana T, Saha G, Rahman M (2008) Effects of discrete isoflux heat source size and angle of inclination on natural convection heat transfer flow inside a sinusoidal corrugated enclosure. Int Commun Heat Mass Transf 35(10):1288–1296 29. Raisi A (2016) Natural convection of non-Newtonian fluids in a square cavity with a localized heat source. Strojnisˇki vestnik J Mech Eng 62(10):553–564 30. Patankar S (1980) Numerical heat transfer and fluid flow. CRC Press, Boca Raton 31. Partal P, Franco JM (2010) Non-Newtonian fluids. Rheology: encyclopaedia of life support systems (EOLSS). UNESCO Eolss, Oxford, pp 96–119
123
